Quantum procedures for approximate optimization by quenching

ABSTRACT

In this disclosure, example quantum algorithms for approximate optimization based on a sudden quench of a Hamiltonian. While the algorithm is general, it is analyzed in this disclosure in the specific context of MAX-EK-LIN2, for both even and odd K. It is to be understood, however, that the algorithm can be generalized to other contexts. A duality can be found: roughly, either the algorithm provides some nontrivial improvement over random or there exist many solutions which are significantly worse than random. A classical approximation algorithm is then analyzed and a similar duality is found, though the quantum algorithm provides additional guarantees in certain cases.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims the benefit of U.S. Provisional Application No.62/782,254 entitled “QUANTUM PROCEDURES FOR APPROXIMATE OPTIMIZATION BYQUENCHING” filed on Dec. 19, 2018, which is hereby incorporated hereinby reference in its entirety.

SUMMARY

In this disclosure, example quantum algorithms for approximateoptimization based on a sudden quench of a Hamiltonian. While thealgorithm is general, it is analyzed in this disclosure in the specificcontext of MAX-EK-LIN2, for both even and odd K. It is to be understood,however, that the algorithm can be generalized to other contexts. Aduality can be found: roughly, either the algorithm provides somenontrivial improvement over random or there exist many solutions whichare significantly worse than random. A classical approximation algorithmis then analyzed and a similar duality is found, though the quantumalgorithm provides additional guarantees in certain cases.

The embodiments disclosed herein include example methods for performingan approximate optimization technique using a quantum quench algorithm.In one embodiment, for instance, a quantum computing device isconfigured to perform an approximate optimization technique toapproximate a solution to a combinatorial optimization problem. Theapproximate optimization technique is then performed on the quantumcomputing device. In this embodiment, the approximate optimizationtechnique includes using a quantum quench algorithm. In certainimplementations, the quench algorithm includes averaging state valuesover a plurality of times. In particular implementations, thecontrolling comprises changing coupling constants without using a jumpor slow change in the coupling constants. In some examples, the changingof the coupling constants is performed non-adiabatically. In certainexamples, the changing of the coupling constants is followed by anequilibration time. In some implementations, the method furthercomprises reading out results of the approximate optimization techniquefrom the quantum computing device; and storing the results in aclassical computing device.

Also disclosed are example embodiments for performing a quantum quenchalgorithm on a classical computing device using simulation of a quantumHamiltonian to perform approximate optimization.

The foregoing and other objects, features, and advantages of thedisclosed technology will become more apparent from the followingdetailed description, which proceeds with reference to the accompanyingfigures.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 illustrates a generalized example of a suitable classicalcomputing environment in which several of the described embodiments canbe implemented.

FIG. 2 is an example of a possible network topology (e.g., aclient-server network) for implementing a system according to thedisclosed technology.

FIG. 3 is another example of a possible network topology (e.g., adistributed computing environment) for implementing a system accordingto the disclosed technology.

FIG. 4 is an exemplary system for implementing the disclosed technology.

FIG. 5 is an example method for performing an approximate optimizationtechnique using a quantum quench algorithm as disclosed herein.

FIG. 6 is another example method for performing an approximateoptimization technique using a quantum quench algorithm as disclosedherein.

DETAILED DESCRIPTION I. General Considerations

As used in this application, the singular forms “a,” “an,” and “the”include the plural forms unless the context clearly dictates otherwise.Additionally, the term “includes” means “comprises.” Further, the term“coupled” does not exclude the presence of intermediate elements betweenthe coupled items. Further, as used herein, the term “and/or” means anyone item or combination of any items in the phrase.

Although the operations of some of the disclosed methods are describedin a particular, sequential order for convenient presentation, it shouldbe understood that this manner of description encompasses rearrangement,unless a particular ordering is required by specific language set forthbelow. For example, operations described sequentially may in some casesbe rearranged or performed concurrently. Moreover, for the sake ofsimplicity, the attached figures may not show the various ways in whichthe disclosed systems, methods, and apparatus can be used in conjunctionwith other systems, methods, and apparatus. Additionally, thedescription sometimes uses terms like “produce” and “provide” todescribe the disclosed methods. These terms are high-level abstractionsof the actual operations that are performed. The actual operations thatcorrespond to these terms will vary depending on the particularimplementation and are readily discernible by one of ordinary skill inthe art.

II. Introduction

For many combinatorial optimization problems, one expects that it is notpossible to obtain an exact solution in polynomial time. Instead, thebest that one can hope for is to obtain an approximate solution. In thisdisclosure, a general approach is presented to constructing quantumapproximation algorithms based on the idea of a quench: a sudden changein the Hamiltonian. Further, a duality is analyzed that one can call“pretty good or very bad”, in which the algorithm either finds anontrivial improvement over random (termed “pretty good”) or there existmany solutions which are significantly worse (termed “very bad”). Asimilar duality in the context of a classical approximation algorithm isalso analyzed.

In this disclosure, the optimization problem MAX-EK-LIN-2 is considered,with the assumption of a degree bound explained below. Roughly speaking,this problem MAX-≤K-LIN-2 considers an objective function which is a sumof terms of degree K; a more precise definition is given later. Aninstance is considered with degree D, so that each degree of freedomparticipates in D terms in the problem. Previous work has shown that forodd K it is possible to obtain a nontrivial approximation of order1/√{square root over (D)} for MAX-EK-LIN-2 using a classical algorithm(see. e.g., B. Barak, A. Moitra, R. O'Donnell, P. Raghavendra, O. Regev,D. Sterner, L. Trevisan, A. Vijayaraghavan, D. Witmer, and J. Wright,arXiv preprint arXiv:1505.03424 (2015)) (initially a quantum algorithmwas found providing weaker approximation guarantees but later theclassical algorithm was discovered (see, e.g., E. Farhi, J. Goldstone,and S. Gutmann, arXiv preprint arXiv:1412.6062 2, 13)). but later aclassical algorithm was discovered with better approximationguarantees). Further, for arbitrary K, the classical algorithm finds asolution which is either better than random by an amount 1/√{square rootover (D)} or worse by an amount of order 1/√{square root over (D)}.(This result for K odd implies the order 1/√{square root over (D)}improvement since if the algorithm find a result result than random byorder 1/√{square root over (D)}, one can change the sign of allvariables an obtain an improvement by order 1/√{square root over (D)}.

Also considered is a slightly different but related classicalapproximation algorithm where it was found (for arbitrary K, though theresult is most interesting for even K) that the duality abovegeneralizes to this: rather than being better or worse by 1/√{squareroot over (D)}, one can instead choose it to be slightly better or muchworse. In this case, if the instance does not have a solution which ismuch worse, then the algorithm is guaranteed to find a slightimprovement.

Embodiments of the disclosed technology also a quantum algorithm basedon quenches. Rather than slowly changing a Hamiltonian as in theadiabatic algorithm (see, e.g., E. Farhi, J. Goldstone, S. Gutmann, J.Lapan, A. Lundgren, and D. Preda, Science 292, 472 (2001) which ingeneral is expected to have trouble with small energy gaps (see. e.g.,B. Altshuler, H. Krovi, and J. Roland, Proceedings of the NationalAcademy of Sciences 107, 12446 (2010))), one can suddenly change theHamiltonian, but then spend some time evolving under the newHamiltonian. In accordance with certain embodiments, a general methodfor approximate optimization is disclosed. In particular examples, thegeneral method is analyzed in the context of MX-EK-LIN-2. Here, asimilar duality is found but with slightly different requirements: insome ways, the quantum algorithm requires only weaker assumptions inorder to find a nontrivial improvement.

A. Problem Definition and Examples

Consider the problem MAX-EK-LIN-2. There are N degrees of freedom, eachof may take values in {−1, +1}. The objective function, which one candenote as H_(Z), is taken to be a weighted sum of monomials of degree atleast 1 and at most K in these variables, e.g., each monomial is aproduct of at least 1 and at most K distinct variables. One can requirethat the weight of each monomial be chosen from {−1, +1}, and that allmonomials be distinct from each other.

Consider an optimization problem where the goal is to maximize thisobjective function. One can emphasize this since a Hamiltonian whichincludes a term proportional to H_(Z), will be later considered andstates near the highest energy state of that Hamiltonian, rather thanthe lowest energy state as more commonly done in physics, will also beconsidered.

One can write the variables as Z_(i) when i ∈{1 . . . N} so that thereare N variables, so for MAX-E2-LIN-2 one has

$\begin{matrix}{{H_{Z} = {\sum\limits_{i < j}{J_{ij}Z_{i}Z_{j}}}},} & (1)\end{matrix}$where J_(ij) is a matrix with entries chosen from {−1, 0, +1}.

A degree bound D will be assumed, so that each variable Z_(i) appears inat most D distinct monomials in H_(Z). Indeed, for simplicity only thecase where each Z_(i) appears in exactly D monomials in H_(Z) will beconsidered. One can define N_(T) to equal the number of terms in H_(Z)so that if every term has degree exactly D (e.g., for MAX-EK-LIN-2) onehas

${N_{T} = \frac{DN}{K}},$and for MAX-K-LIN-2 one has

$\begin{matrix}{{DN} \geq N_{T} \geq {\frac{DN}{K}.}} & (2)\end{matrix}$

A random assignment has expectation value of H_(Z) equal to 0. Typicallyin computer science, one regards each of these monomials as aconstraint: the constraint is satisfied if the monomial is equal to +1and it is violated otherwise, so that the number of satisfiedcoustraints is equal to the value of H_(Z) plus N_(T)/2. Hence, randomassignment satisfies half the constraints on average. Then, theapproximation ratio achieved by some assignment to the variables isdefined to be the fraction of constraints satisfied by that assignmentdivided by the fraction of constraints satisfied by the optimalassignment.

One can define the approximation ratio differently: define it to be thevalue of H_(Z) for a given assignment divided by the value of H_(Z) inthe optimal assignment. That is, this term N_(T)/2 will not be added.

In certain examples, it is said that an assignment improves by a factorƒ over random if it has H_(Z)≥ƒN_(T). Further, in certain examples, itis said that an algorithm is worse than random by a factor ƒ if it hasH_(Z)≤−ƒN_(T).

It is known that it is always possible to obtain an Ω(1/D) approximationratio with a polynomial time algorithm for MAX-K-LIN-2. (see. e.g., J.Hastad, Information Processing Letters 74, 1 (2000)). More strongly, forany instance it is always possible to find an assignment which improvesby a factor 1/D over random in polynomial expected time. For odd K,however, it is possible to improve over a random assignment byexp(−O(K))/√{square root over (D)} in polynomial time. (See, e.g., B.Barak, A. Moitra, R. O'Donnell, P. Raghavendra, O. Regev, D. Sleurer, L.Trevisan, A. Vijayaraghavan, D. Witmer, and J. Wright, arXiv preprintarXiv:1505.03424 (2015).)

One cannot expect to have such an improvement for even K simply becausethere exist families of instances in which no assignment has H_(Z)larger than N_(T)·

(1/D). For K=2, a simple such example is to choose

$\begin{matrix}{H_{Z} = {- {\sum\limits_{i < j}{Z_{i}{Z_{j}.}}}}} & (3)\end{matrix}$Here one has taken D=N−1 so that every variable is in some monomial withanother variable. It is possible to obtain a very large negativeexpectation value of H_(Z) (e.g., −N(N−1)/2) by choosing all Z_(i) tohave the same sign, but for even N, the maximum positive expectationvalue of H_(Z) is to choose N/2 tat the Z_(i) to equal +1 and theremainder to equal −1, giving expectation value N/2, which isproportional to N_(T)/N. This example provides an early example of theduality: the maximum improvement over random is quite small (O(1/D)) butone can find an assignment is a factor Ω(1) worse than random.

For K=2m, one can generalize example (3) to give an instance forMAX-EK-LIN-2 as follows: let N=mD. Divide the set of mD variables into Ddisjoint sets, each containing m variables. Label the sets by integersin 1, . . . , D. Let {tilde over (Z)}_(i) be the product of thevariables in the i-th set. Let H_(Z)=−Σ_(i<j){tilde over (Z)}_(i){tildeover (Z)}_(j).

B. Outline, Notation, and Results

In section III, an example embodiment of the quench algorithm isdefined, both in the specific form that is analyzed later as well assome variants that may be useful. In section IV, some results arecollected that will be useful in analyzing the classical algorithm thatare given later as well as in analyzing the quantum algorithm. Insection V, the classical algorithm is defined and analyzed; this sectionwill give some notation and probabilistic results that will be used inthe quantum algorithm. In section VI, the example embodiment of thequantum algorithm is analyzed.

III. Quench Algorithm

To define the algorithm, one can promote the variables to qubits, andlet Z_(i) be the Pauli Z operator on the i-th qubit. Let X_(i) be thePauli N operator on the i-th qubit and let

$\begin{matrix}{X = {\sum\limits_{i}{X_{i}.}}} & (4)\end{matrix}$

The following algorithm can then be used. Let

$\begin{matrix}{{H = {X + {\frac{\alpha}{D}H_{Z}}}},} & (5)\end{matrix}$

where α is a scalar chosen later. The system can be prepared in thestate ψ₊ maximally polarized in the + direction so that X_(i)=+1 for alli. One can then evolve the system under Hamiltonian H for a time T thatwill be explained later. This time will in all cases be at most poly(N);indeed, the analysis will be for T=O(1). Hence, this evolution can beperformed in polynomial time on a quantum computer in time polynomial int_(max) and polynomial in the inverse error (see, e.g., D. W. Berry, A.M. Childs, and R. Kothari, 2015 IEEE Symposium on Foundations ofComputer Science, 792 (2015)); indeed, the simulation can be performedin time polylogarithmic in the inverse error but this will not beneeded. In any simulation algorithm on a quantum computer, one candiscretize the variable t; for example, one may choose it to equal aninteger multiple of some time t_(min) for some t_(min) which ispolynomially small; this causes only a polynomially small error.Finally, one can measure the state of the system in the computationalbasis, giving an assignment of variables Z_(i).

In the analysis of the algorithm, one can ignore all the errorsassociated with the time evolution and the discretization of time, sincea polynomially small error is negligible as may be verified.

When one applies this algorithm, one can often repeat the algorithmseveral times with T chosen from an appropriate distribution asdescribed in section VI, repeating the H_(Z) measurement each time. Inthis regard, it is interesting to think about the state arising fromaveraging T over an interval of times; by choosing the time from arandom distribution (or more generally, by performing phase estimationof the Hamiltonian H) one can decohere the system in an eigenbasis. Thefixed evolution has a similar effect but is easier to analyze using thetechniques here. Also, it may be useful to consider a generalization ofthe algorithm in which one does some slow (but not necessarilyadiabatic) evolution of the Hamiltonian from an initial Hamiltonian X toH=X+(α/D)H_(Z), followed by an additional time evolving underH=X+(α/D)H_(Z).

A. Motivation

In this section, the algorithm is heuristically explained. The timeevolution has two purposes. The first is to decohere differenteigenstates of the Hamiltonian as mentioned; for a fixed time, theevolution still for time t still produces a pure state, but stillproduces some change in phase for different energies which has a similareffect to a random evolution. The second purpose is to do it in a waythat conserves energy. One hopes that the decoherence between differenteigenstates will lead to a reduction in the expectation value of X,since one hopes that individual eigenstates will not have large X. Thisreduction will lead to a positive expectation value of H_(Z) due to theenergy conservation as will now be explained: this energy conservationis the second reason for the time evolution.

For arbitrary operators O, H and scalar t, define τ_(t)^(H)(O)=exp(itH)O exp(−itH). Define

O

₊≡

ψ₊|O|ψ ₊

.   (6)One has

τ_(T) ^(H)(H)

₊=

H

₊ =N,   (7)Independent of T by energy conservation. Hence, one has

$\begin{matrix}{\left\langle {\tau_{T}^{H}\left( H_{Z} \right)} \right\rangle = {D{\frac{N - \left\langle {\tau_{T}^{H}(X)} \right\rangle_{+}}{\alpha}.}}} & (8)\end{matrix}$That is, if the state at time T has an expectation value of X that issmaller than the maximal (e.g. smaller than N), it necessarily has anexpectation value of H_(Z) that is positive. In other words, it hasobtained some solution that is better than random. This is one of thenotable aspects behind the quench algorithm.

B. Heuristic Choices of α

This section discusses now to choose α. A calculation is given thatintroduces some of the notation used later. Perturbation theory to onlya second order is considered, and a purely heuristic treatment of higherorders to motivate the choice of α is given. Later, a differenttreatment is given.

Consider the series for τ_(T) ^(H)(X_(i)) for some given i: for anyoperator O, one has the series

$\begin{matrix}{{\tau_{T}^{H}(O)} = {O - {{iT}\left\lbrack {O,H} \right\rbrack} - {\frac{T^{2}}{2}\left\lbrack {\left\lbrack {O,H} \right\rbrack,H} \right\rbrack} + {i{\frac{T^{3}}{3!}\left\lbrack {\left\lbrack {\left\lbrack {O,H} \right\rbrack,H} \right\rbrack,H} \right\rbrack}} + {\frac{T^{4}}{4!}\left\lbrack {\left\lbrack {\left\lbrack {\left\lbrack {O,H} \right\rbrack,H} \right\rbrack,H} \right\rbrack,H} \right\rbrack} + \ldots}} & (9)\end{matrix}$

So, one has

$\begin{matrix}{{{\tau_{T}^{H}(X)} = {X - {i{\frac{\alpha\; T}{D}\left\lbrack {X,H_{Z}} \right\rbrack}} - {\frac{\alpha^{2}}{D^{2}}{\frac{T^{2}}{2}\left\lbrack {\left\lbrack {X,H_{Z}} \right\rbrack,H} \right\rbrack}} + \ldots}}\;,} & (10)\end{matrix}$where the dots denote terms of order T³ or higher. Hence,

$\begin{matrix}\begin{matrix}{\left\langle X \right\rangle = {\left\langle {X - {\frac{\alpha^{2}}{D^{2}}{\frac{T^{2}}{2}\left\lbrack {\left\lbrack {X,H_{Z}} \right\rbrack,H} \right\rbrack}}} \right\rangle_{+} + \ldots}} \\{{= {N - {\frac{\alpha^{2}}{D^{2}}\frac{T^{2}}{2}N} + \ldots}}\;,}\end{matrix} & (11)\end{matrix}$where the following is use

[[X, H_(Z)], H]

₊=Σ_(i)

[[X_(i), H_(Z)], H_(Z)]

₊=ND. and so

$\begin{matrix}{\left\langle H_{Z} \right\rangle = {{\frac{\alpha\; T^{2}}{2}N} + \ldots}} & (12)\end{matrix}$

Of course, the higher order corrections to this perturbation theory mustbecome important for large enough T, α. For one thing, once T≲1, theeffects of higher order terms in TX in the exponential become important,e.g., one must consider higher order commutators such as [[[[H_(Z)], X],X], H_(Z)]. However, one might hope that for some T of order unity (forexample T=½) the ignore higher orders in TX will not be too important;maybe they will not be negligible but one can hope that they will onlyslightly reduce the result.

However, even for such a fixed T=½, one cannot ignore higher order termsin (α/D)H_(Z) for large enough α. For example, if α is sufficientlylarger than √{square root over (D)}, one would find that Eq. (11) givesa result for X which is smaller than −N, which is impossible.

So, the most optimistic outcome that one can hope for is that secondorder perturbation theory is roughly accurate up to some T of orderunity such as T=½ and up to α proportional to √{square root over (D)}.If so, one would find that the best choice of α would be to take αproportional to √{square root over (D)}, in which case one would have

H_(Z)

proportional to N√{square root over (D)}, which is proportional toN_(T)·Ω(1/√{square root over (D)}).

However, this heuristic analysis is likely too optimistic. Suchsolutions do exist for MAX-EK-LIN-2 for odd K (though it has not beenshown that the algorithm finds them), but they do not exist in general,such as the example of (3).

IV. Rounding

In this section, some general results are given on how, given a solutionto an optimization problem for a polynomial in several vectorialvariables, one can construct a solution to the same problem where allvariables are chosen to be the same. Theorem 1 is the main result. Thisresult will be used in both the classical and quantum algorithms; thevectors {right arrow over (w)}_(a) are the solution to the problem usingseveral vectorial variables. while the {right arrow over (u)} is thesolution with all variables the same.

The way one can use these theorems is as follows. H_(Z) as a degree-Kpolynomial in variables Z_(i) is given. Each Z_(i) is chosen from {−1,+1}. Let {right arrow over (Z)} be a vector of choices of variables Z.One can write H_(Z)({right arrow over (Z)}) to denote the value of H_(Z)for that given set of choices.

One can randomly round choices of Z_(i) from the interval [−1, +1] tochoices of Z_(i) from the discrete set Z_(i)={−1, +1} while preservingexpectation value. Formally, consider a vectorial variable {right arrowover (v)} with each entry is chosen from the interval [−1, −1]. Then,independently choosing each Z_(i) at random from {−1, +1}, picking theprobability for each Z_(i) so that

[Z_(i)]=v_(i), one has

[H_(Z)({right arrow over (Z)})]=H_(Z)({right arrow over (v)}).

Now, let one define a polynomial H_(Z)({right arrow over (v)}₁, {rightarrow over (v)}₂, . . . , {right arrow over (v)}_(K)) which depends uponK different vectorial variables as follows. This polynomial will behomogeneous of degree 1 in each variable. For each term in H_(Z) of theform cZ_(i) ₁ , Z_(i) ₂ , . . . Z_(i) _(K) , where c is a scalar and i₁,i₂, . . . , i_(K) are a sequence of distinct choices of i, one has acorresponding term in H_(Z)({right arrow over (v)}₁, . . . , {rightarrow over (v)}_(K)) equal to

${c\frac{1}{K!}{\sum\limits_{\pi}{\left( {\overset{\rightarrow}{\upsilon}}_{1} \right)_{i_{\pi{(1)}}}\left( {\overset{\rightarrow}{\upsilon}}_{2} \right)_{i_{\pi{(1)}}}{\ldots\left( {\overset{\rightarrow}{\upsilon}}_{K} \right)}_{i_{\pi{(1)}}}}}},$where the sum is over permutations π on K elements and ({right arrowover (v)}_(a))_(b) denotes the b-th entry of vector {right arrow over(v)}_(a). For example, for K=2, given a term −Z₂Z₃ one has thecorresponding term −(½)({right arrow over (v)}₁)₂({right arrow over(v)}₂)₃−(½)({right arrow over (v)}₂)₃({right arrow over (v)}₁)₂. Here inan abuse of notation, the same symbol H_(Z)(·) is used for two differentfunctions, one depending on K vectorial arguments and one depending on asingle vectorial argument.

Note thatH _(Z)({right arrow over (v)}, {right arrow over (v)}, . . . , {rightarrow over (v)})=H _(Z)({right arrow over (v)}).   (13)The purpose of the rounding results will be, given some choice of {rightarrow over (v)}₁, . . . , {right arrow over (v)}_(K)) such thatH_(Z)({right arrow over (v)}₁, . . . , {right arrow over (v)}_(K)) has acertain magnitude, one can find a choice of {right arrow over (v)} suchthat H_(Z)({right arrow over (v)}) obeys certain conditions on itsmagnitude.

This will then be used in the classical setting in the following simpleway: one can pick some vector {right arrow over (w)}₂ at random and thenchoose {right arrow over (w)}₁ greedily to optimize H_(Z)({right arrowover (w)}₁, {right arrow over (w)}₂, {right arrow over (w)}₂, {rightarrow over (w)}₂, . . . , {right arrow over (w)}₂). Here the variable{right arrow over (w)}₁ appears 1 time while the variable {right arrowover (w)}₂ appears K−1. This will give one the choice of K differentvectorial variables (though one variable is repeated K−1 times) fromwhich one will construct a solution with a single variable.

Case 3 of the following theorem will be useful for the classicalalgorithm. Case 2 follows from case 3 with ϵ=1 but has slightly tighterbounds. Case 1 is given for completeness. Thus, the reader may consideronly case 3.

Theorem 1. Let P({right arrow over (v)}₁, {right arrow over (v)}₂, . . ., {right arrow over (v)}_(K)) be a polynomial in vectorial variables{right arrow over (v)}₁, . . . , {right arrow over (v)}_(K) which ishomogeneous of degree 1 in each argument so that

$\begin{matrix}{{{P\left( {{\overset{\rightarrow}{\upsilon}}_{1},\ldots\;,{\overset{\rightarrow}{\upsilon}}_{K}} \right)} = {\sum\limits_{i_{1},\;\ldots\;,\; i_{K}}{a_{i_{1},\;\ldots\;,\; i_{K}}{\prod\limits_{a}\;\left( {\overset{\rightarrow}{\upsilon}}_{a} \right)_{i_{a}}}}}},} & (14)\end{matrix}$where ({right arrow over (v)}_(a))_(i) denotes the i-th entry of vector{right arrow over (v)}_(a).

Assume that all vectors {right arrow over (v)}_(a) have the same numberof entries, and assume that P is symmetric under permuting itsarguments, e.g., that a_(i) _(1, . . . ,) _(i) _(K) is symmetric underpermuting its arguments.

Then the following holds:

-   -   1. Suppose that there exist some vectors {right arrow over        (w)}₁, . . . , {right arrow over (w)}_(K) such that P({right        arrow over (w)}₁, . . . , {right arrow over (w)}_(K))=C and such        that |{right arrow over (w)}_(a))_(i)|≤1 for all a, i. Then,        there exits some vector {right arrow over (u)} with        |({right arrow over (u)}_(i))|≤1        for all i such that

$\begin{matrix}{{{P\left( {\overset{\rightarrow}{u},\overset{\rightarrow}{u},\ldots\;,\overset{\rightarrow}{u}} \right)}} \geq {\frac{K!}{K^{K}}{C.}}} & (15)\end{matrix}$

-   -   2. Suppose that there exist vectors {right arrow over (w)}₁,        {right arrow over (w)}₂ such that P({right arrow over (w)}₁,        {right arrow over (w)}₂, {right arrow over (w)}₂, {right arrow        over (w)}₂, . . . , {right arrow over (w)}₂)=C and such that        |({right arrow over (w)}_(a))_(i)|≤1 for all a, i. (That is, the        variable {right arrow over (w)}₁ appears 1 time while the        variable {right arrow over (w)}₂ appears K−1 times. Then, there        exists some vector {right arrow over (u)} with        |({right arrow over (u)}_(i))|≤|({right arrow over        (w)}₁)_(i)|+|({right arrow over (w)}₂)|_(i)        for all i such that        |P({right arrow over (u)}, {right arrow over (u)}, . . . ,        {right arrow over (u)})|≥P({right arrow over (w)} ₂ , {right        arrow over (w)} ₂ , . . . , {right arrow over (w)} ₂)+C·Ω(1/K).          (16)    -   3. Suppose that there exist vectors {right arrow over (w)}₁,        {right arrow over (w)}₂ such that P({right arrow over (w)}₁,        {right arrow over (w)}₂, {right arrow over (w)}₂, {right arrow        over (w)}₂, . . . , {right arrow over (w)}₂)=C. Then for any        ϵ>0, at least one of the following two possibilities holds:        -   A there exists some vector {right arrow over (u)} with            |({right arrow over (u)}_(i))|≤|({right arrow over            (w)}₁)_(i)|+|({right arrow over (w)}₂)|_(i) for all i such            that            |P({right arrow over (u)}, {right arrow over (u)}, . . . ,            {right arrow over (u)})|≥P({right arrow over (w)} ₂ , {right            arrow over (w)} ₂ , . . . , {right arrow over (w)}            ₂)+ϵC·Ω(1)   (17)            or        -   B then exists some vector {right arrow over (u)} with            |({right arrow over (u)}_(i))|≤|({right arrow over            (w)}₁)_(i)|+|({right arrow over (w)}₂)|_(i) for all i such            that            |P({right arrow over (u)}, {right arrow over (u)}, . . . ,            {right arrow over (u)})|≤P({right arrow over (w)} ₂ , {right            arrow over (w)} ₂ , . . . , {right arrow over (w)}            ₂)−C·exp(−O(K))/ϵ.   (18)

Further, in all cases, one can find {right arrow over (u)} up to anydesired nonzero error in a time linear in N, exponential in K, and atmost polynomial in inverse error compared to magnitude of the terms inthe polynomial.

Note that item 1 above allows all of the {right arrow over (w)}_(a) tobe distinct. Items 2,3 consider the case of just two different {rightarrow over (w)}_(a), with {right arrow over (w)}₂ repeated K−1 times inthe argument of P(·). One can summarize item 2 as saying that one canobtain a solution whose absolute value is close to C, while item 3 canbe summarized for small ϵ as saying that, compared to P({right arrowover (w)}₂, {right arrow over (w)}₂, . . . , {right arrow over (w)}₂),either one can improve by a small amount (this is the “pretty good”) orthere is a solution which is much worse (this is the “very bad”). Notealso that the bound on |({right arrow over (u)})_(i)| is different initem 2 compared to items 1,3.

One can now prove the theorem. Define a function {right arrow over(u)}(·), from

^(K) to vectors, by

$\begin{matrix}{{{\overset{\rightarrow}{u}\left( {x_{1},\ldots\;,x_{K}} \right)} = {\sum\limits_{a}{u_{a}{\overset{\rightarrow}{\upsilon}}_{a}}}},} & (19)\end{matrix}$where x_(a){right arrow over (v)}_(a) denotes the vector with i-th entryequal to x_(a)({right arrow over (v)}_(a))_(i).

One can first prove item 1 of theorem 1. One needs:

Lemma 1. Let p(x₁, . . . , x_(K)) be a polynomial (not necessarilyhomogenous) of degree at most K in real variables x₁. . . , x_(K).Suppose that the coefficient of the term Π_(i)x_(i) in p(·) is equal toC. Then, for some choice of x₁, . . . , x_(K)∈{−1, +1}^(K) one has that|p(x₁, . . . , x_(K))|≥C.Proof. It is claimed that

$\begin{matrix}{C = {\frac{1}{2^{K}}{\sum\limits_{x_{1},\;\ldots\;,\;{x_{K} \in {\{{{- 1},{+ 1}}\}}^{K}}}{\left( {\prod\limits_{i}\; x_{i}} \right) \cdot {{p\left( {x_{1},\ldots\;,x_{K}} \right)}.}}}}} & (20)\end{matrix}$This holds because any term in p(·) proportional to Π_(i) x_(i) ^(d)^(i) for some sequence of integers d_(i) will vanish in the weighted sumabove unless all d_(i) are odd. However, since p(·) has degree d, theonly such nonvanishing term is that with all d_(i)=1.

Hence, |C|≤max_(x) ₁ _(, . . . , x) _(K) _(∈{−1, +1}) _(K) (|p(x₁, . . ., x_(K))|). □

To prove item 1, consider polynomial Q(x₁, . . . , q_(K))≡P({right arrowover (u)}(x₁, . . . , x_(K)), . . . , {right arrow over (u)}(x₁, . . . ,x_(K))). The polynominal Q(·) is of degree K and the coefficient ofΠ_(i) x_(i) in Q(·) is equal to CK!. So, by lemma 1, there exists somechoice of x₁. . . , x_(K) ∈{−1, 1}^(K) such that |Q(x₁, . . . ,x_(K))|≥CK!. Set

$\overset{\rightarrow}{u} = {\frac{1}{K}\left. \overset{\rightarrow}{(}{x_{1},\ldots\;,x_{K}} \right)}$so that indeed |({right arrow over (u)})_(i))|≤1 for all i.

Then, |P({right arrow over (u)}, . . . , {right arrow over(u)})|≥(1/K)^(K) CK!. This prove item 1 and trivially one can find thechoice of {right arrow over (u)} by iterating over the 2^(K) possiblechoices of x₁, . . . ,x_(K) ∈{−1, 1}^(K).

Item 2 will next be proved. The following lemma applies:

Lemma 2. Let p(x) be a polynomial of degree K withp(x)=Σ_(0≤i≤d)a_(i)x^(i). Then, for K oddmin_(x∈[−1,1])(|(p(x)|)≥|a ₁ |/K,   (21)and for K evenmin_(x∈[−1,1])(|(p(x)|)≥|a ₁|/(K−1),   (22)Proof. The proof is similar to the proof that that the Chebyshevpolynomials have minimum absolute value on the interval [−1, 1] amongall polynomials with given leading coefficients, e.g., with given valueof a_(K). In this case, one can instead fix the value of a₁, but theproof is almost the same.

First, without loss of generality, one can assume that p(x)=−p(−x), as(p(x)−p(−x))/2 is also a polynomial of degree K with coefficient of thelinear term also equal to a₁ and |(p(x)−p(−x))/2|≤max(|p(x)|, |p(−x)|).So, one can assume that K is odd and the result for even K will followimmediately from the result for odd K.

Also, without loss of generality one may assume that a₁=1. Indeed, ifa₁=0, then the result is trivial true, while for any nonzero a₁, one caninstead consider p(x)/a₁.

Assume that the lemma is false, e.g., assume that p(x) has maximumabsolute value on the interval [−1, 1] which is strictly smaller than1/K.

Let T_(n)(x) be the Chebyshev polynomials of first kind. For odd K,−(−1)^(K)·T_(K)(x)/K is a polynomial of degree K which has coefficientof the linear term equal to 1. Further, −(−1)^(K)·T_(K)(x)/K has amaximum absolute value on the interval [−1, 1] equal to 1/K and itattains this maximum K+1 times on this interval at points x=cos(kπ/K)for 0≤k≤K. Let q(x)=p(x)+(−1)^(K)·T_(K)(x)/K. So, q(x) has coefficientof the linear term equal to zero, e.g., since it is an odd function ofx, one has q(x)=Σ_(i=3,5, . . . ,K)b^(i)x^(i) for some coefficientsb_(i) and further by the assumption that p(x) has absolute valuestrictly smaller than 1/K on the interval, one has that at pointsx=cos(kπ/K) the sign of q(x) is the same as the sign of(−1)^(K)·T_(K)(x)/K. So, since the sign of T_(K)(x) alternates at thesepoints, e.g., the sign for even k is opposite to that for odd k, one hasthat q(x) changes sign at least K times so q(x) must have at least K−1distinct zeros. However, q(x) has degree K and the root at x=0 is triplydegenerate so in fact q(x) can only have at most K−2 distinct zeros,giving a contradiction. □

Define polynomial Q(x)≡P({right arrow over (u)}(x, 1, 1, . . . , 1), . .. . , {right arrow over (u)}(x, 1, 1, . . . , 1)), e.g., in the argumentof {right arrow over (u)}, 1 is repeated a total of K−1 times. Applyinglemma 2 to p(x)=Q(x), the result follows. One can find an x whichmaximizes |Q(x)| up to any given error by exhaustively trying a discreteset of points on the interval [−1, 1] with the spacing between pointsdependent on the error.

Finally, item 3 is proven. The following lemma is applied:

Lemma 3. Let p(x)=Σ_(0≤i≤d)a_(i)x^(i) be a degree-d polynomial in realvariable x. Let p_(max)=max_(x∈[−1,1])p(x). Leta_(max)=max_(i≥1)|a_(i)|. Thenp_(max) ≥a ₀+(⅙)a ₁ ² /a _(max)   (23)

Remark: the factor ⅙ in the above equation is not optimal. It can betightened easily. Indeed, for a₁<<a_(max), the factor ⅙ approaches ½.

Proof Consider p(x₀) for x₀=a₁/4a_(max)). One hasp(x₀)=a₀+(¼)a_(max)+Σ_(1≤i≤d)a_(i)x_(i) ^(i). So,

$\begin{matrix}{{p\left( x_{0} \right)} \geq {a_{0} + {\left( {1/4} \right){a_{1}^{2}/a_{\max}}} - {{❘{{\sum\limits_{{2i} \leq i \leq \infty}{a_{\max}\left( {{❘a_{1}❘}/\left( {4a_{\max}} \right)} \right)}^{i}} \geq {a_{0} + {\left( {1/4} \right){a_{1}^{2}/a_{\max}}} -}}❘}{\sum\limits_{{2i} \leq i \leq \infty}{{a_{\max}\left( {{❘a_{1}❘}/\left( {4a_{\max}} \right)} \right)}^{2}\left( {{❘a_{i}❘}/\left( {4a_{\max}} \right)} \right)^{i - 2}}}}} \geq {a_{0} + {\left( {1/4} \right){a_{1}^{2}/a_{\max}}} - {{a_{\max}\left( {{❘a_{1}❘}/\left( {4a_{\max}} \right)} \right)}^{2}{\sum\limits_{{2i} \leq i \leq \infty}\left( {1/4} \right)^{i - 2}}}} \geq {a_{0} + {\left( {1/6} \right){a_{1}^{2}/{a_{\max}.}}}}} & (24)\end{matrix}$

Define polynomial Q(x)≡P({right arrow over (u)}(x, 1, 1, . . . , 1), . .. , {right arrow over (u)}(x, 1, 1, . . . , 1)) as above. Apply lemma 3with a₁=C. If item A of theorem 1 does not hold for some given ϵ, then(⅙)C²/a_(max)<ϵC so a_(max)>(⅙)(C/ϵ). So for some i≥1, |a_(i)|>(⅙)(C/ϵ).So,

${{❘a_{i}❘} = {{\begin{pmatrix}K \\i\end{pmatrix}{❘{P\left( {{\overset{\rightarrow}{w}}_{1},\ldots,{\overset{\rightarrow}{w}}_{1},{\overset{\rightarrow}{w}}_{2},\ldots,{\overset{\rightarrow}{w}}_{2}} \right)}❘}} > {\left( {1/6} \right)\left( {C/\epsilon} \right)}}},$where {right arrow over (w)}₁ appears i times in the argument of P(·)and {right arrow over (w)}₂ appears K−i times. So, by item 1 of theorem1, there is some choice of {right arrow over (u)} with |({right arrowover (u)}_(i))|≤1 for all i such that

${❘{P\left( {\overset{\rightarrow}{u},\overset{\rightarrow}{u},\ldots,\overset{\rightarrow}{u}} \right)}❘} \geq {\frac{1}{\begin{pmatrix}K \\i\end{pmatrix}}\frac{K!}{K^{K}}\left( {1/6} \right){\left( {C/\epsilon} \right).}}$Since

${{\frac{1}{\begin{pmatrix}K \\i\end{pmatrix}}\frac{K!}{K^{K}}\left( {1/6} \right)} \geq {\exp\left( {- {O(K)}} \right)}},$the result follows.

This completes the proof.

V. Classical Algorithm

The classical optimization algorithm will now be described.

Some notation that will be useful both here and in the analysis of thequantum algorithm will now be introduced.

Let one define F_(i) (the symbol “F” is for “force”, e.g., a derivativeof energy with respect to some coordinate) to equal Z_(i) times the sumof terms in H_(Z) that include Z_(i). For example, for K=4 andH_(Z)=Z₁Z₂Z₃Z₄+Z₁Z₃Z₄Z₅ then F₁=Z₂Z₂Z₄+Z₃Z₄Z₅. The “force” depends uponthe choice of Z_(i) so one will sometimes write F_(i)({right arrow over(Z)}) to indicate its dependence on {right arrow over (Z)}.

A. Some Probability Bounds

In this section, some probability bounds are collected that will be usedto analyze this algorithm, as well as to analyze the classicalalgorithm.

Algorithm 1 Classical algorithm 1. Fix some real number 0 < p < 1.Choose a set S of degrees of freedom, by including each degree offreedom in S independently with probability p. 2. Define vectorialvariables {right arrow over (w)}₁, {right arrow over (w)}₂ as follows;the index of the vectorial variable will correspond to degrees offreedom. Let {right arrow over (w)}₂ be a vector with ({right arrow over(w)}₂)_(i) = 0 for i ∈ S while for i ∉ S one can choose ({right arrowover (w)}₂)_(i) to +1 or −1 independently and uniformly at random. Onecan choose vector {right arrow over (w)}₁ so that ({right arrow over(w)}₁)_(i) = 0 for i ∈ S while for i ∈ Sone can choose ({right arrowover (w)}₁)_(i) “greedily”. That is, ({right arrow over (w)}₁), = +1 ispicked if F₁ ({right arrow over (w)}₂) > 0 and ({right arrow over(w)}₂)_(i) = −1 otherwise. 3. Finally, apply item 3 of theorem 1. Bythis item, for any ∈ > 0, one can either find a choice of {right arrowover ({right arrow over (u)})} such that H_(Z)({right arrow over (u)}) ≥H_(Z)({right arrow over (w)}₂ ) + ∈C · Ω(1) or such that H_(Z)({rightarrow over (u)}) ≥ H_(Z)({right arrow over (w)}₂) − C · Ω(1)/∈, where C= Σi∈s |F_(i)|.

By theorem 9.23 of R. O'Donnell, Analysis of boolean functions(Cambridge University Press, 2014), for any function ƒ of degree at mostK from {−1, 1}^(N)→

one has for any t≥(2e)^(K/2) that

$\begin{matrix}{{\Pr_{x \in {\{{{- 1},1}\}}^{N}}\left\lbrack {{❘{f(x)}❘} \geq {t{{\mathbb{E}}\left\lbrack {❘f❘}^{2} \right\rbrack}^{1/2}}} \right\rbrack} \leq {{\exp\left( {{- \frac{K}{2e}}t^{2/K}} \right)}.}} & (25)\end{matrix}$

By theorem 9.24 of R. O'Donnell, Analysis of boolean functions(Cambridge University Press, 2014), for any nonconstant function ƒ ofdegree at most K from {−1, 1}^(N)→

,Pr _(x∈{−1,1}) _(N) [ƒ(x)>

[|ƒ|]]≥¼ exp−2K.   (26)

Hence, for any nonconstant function ƒ of degree at most K from {−1,1}^(N)→

, by applying Eq. (26) to ƒ², one hasPr _(x∈{−1,1}) _(N) [|ƒ(x)|>

[|ƒ|²]^(1/2)]≥¼ exp−4K.   (27)

Applying these bounds to the force F_(i), one finds that the average of|F_(i)| is at least √{square root over (D)} exp(−O(K)). At the sametime, the expectation value of H_(Z)({right arrow over (w)}₂) is equalto zero.

B. Analysis of Classical Algorithm

Since |F_(i)|≥√{square root over (D)}exp(−O(K)), one finds that theconstant C in the algorithm has average value at least N√{square rootover (D)} exp(−O(K)).

The algorithm chooses either case 3A or case 3B at least half the time(or any other number Ω(1) rather than one half) and in that case, theaverage of |F_(i)| must still be at least √{square root over(D)}exp(−O(K)). Hence, at least one of the following holds: when thealgorithm chooses case 3A at least half the time and has expectedH_(Z)({right arrow over (u)}) at least Nϵ√{square root over (D)}exp(−O(K)) or the algorithm chooses case 3B at least half the time andhas expected H_(Z)({right arrow over (u)}) at most −N √{square root over(D)} exp(−O(K))/ϵ.

Note that for odd K, one can guarantee then expected H_(Z)({right arrowover (U)})≥√{square root over (D)} exp(−O(K)) as in B. Barak, A. Moitra,R. O'Donnell, P. Raghavendra, O. Regev, D. Steurer, L. Trevisan, A.Vijayaraghavan, D. Witmer, and J. Wright, arXiv preprintarXiv:1505.03424 (2015), since one can pick ϵ=1 and if case 3B occurs,one can change the sign of all variables.

Note also that while the guarantee on the classical algorithm involvesthe expected value of H_(Z), since the expected value is within1/poly(D) of the optimal value (which is O(ND)), by repeating thealgorithm poly(D) times one can, with probability at least one-half,obtain a solution within a constant factor of the expected value.

VI. Analysis of Quantum Algorithm

An embodiment of the quantum quench algorithm disclosed herein will nowbe analyzed in more detail. From Eq. (8),

$\left\langle {\tau_{T}^{H}\left( H_{Z} \right)} \right\rangle = {D{\frac{N - \left\langle {\tau_{T}^{H}(X)} \right\rangle_{+}}{\alpha}.}}$One can estimate

τ_(T) ^(H)(X)

₊,

Consider site i. One can estimate

τ_(T) ^(H)(X_(i))

₊. Summing over i will give

τ_(T) ^(H)(X)

₊.

The basic physical idea is that if one can ignore the time dependence ofthe force F_(i), then one can approximate

τ_(T) ^(H)(X_(i))

₊ by the expectation value of X_(i) assuming that the spin i evolves fora time T under a time-independent Hamiltonian. This time-independentHamiltonian a transverse field of strength 1 (e.g., the term X_(i) inthe Hamiltonian) and to a parallel field (α/D)F_(i), where F_(i) is theforce assuming that all other spins Z_(j) for j≠i are drawn from auniformly random distribution (because at time T=0, the state of thesystem is ψ₊ which has equal amplitude on all states). In this case,similar to the analysis of the classical algorithm before, the forceF_(i) is likely to be at least of order √{square root over (D)} in whichcase one will have 1−

τ_(T) ^(H)(X_(i))

₊˜(α/D)²

F_(i) ²

+T²˜α²T²/D.

However, one cannot always neglect the time dependence of the force. Toestimate whether or not the time dependence of the force is significant,one can compare the time-derivative of the force to √{square root over(D)}/T. If the time-derivative of the force is small enough compared to√{square root over (D)}/T, then the approximation of the above paragraphwill be valid. On the other hand, if the time-derivative is not sosmall, one can derive a similar duality to the classical case. Indeed,one will also have some stronger results here in the case that the finalstate has large expectation value of X.

In subsection VIA, the time-independent case is anayzed. Subsection VICdescribes a toy example where one can see the effects oftime-dependence. In subsection VID, consideration is given to the errorsby ignoring the time-dependence and the main results are obtained.

A. Time-Independent Force

Let one first analyze the time-independent force approximation in moredetail before considering the time-dependence. One wishes to compute

$\left\langle {{\exp\left( {{- {i\left( {{\frac{\alpha}{D}Z_{i}F_{i}} + X_{i}} \right)}}T} \right)}\psi_{+}{❘X_{i}❘}{\exp\left( {{- {i\left( {{\frac{\alpha}{D}Z_{i}F_{i}} + X_{i}} \right)}}T} \right)}{\psi_{+}.}} \right.$That is, consideration is given to an evolution under a Hamiltonianwhich includes the coupling (α/D)Z_(i)F_(i) and the transverse fieldX_(i), but ignoring any other coupling terms which would give theremaining qubits a time-dependence in the Z-basis.

As in the analysis of the classical case, the probability that|F_(i)|≥√{square root over (D)} is at least ¼ exp−4K. At the same time,by Eq. (25), the probability that |F_(i)|≥t√{square root over (D)} fort≥(2e)^(K/2) is at most

${\exp\left( {{- \frac{K}{2e}}t^{2/K}} \right)}.$Picking t sufficiently large (for example, t=C^(K/2) for sufficientlylarge, K-independent constant C suffices), this probability is muchsmaller than ⅛ exp(−4K). So, with probability at least ⅛ exp(−4K), onehas |F_(i)|∈[√{square root over (D)}, C^(K/2) √{square root over (D)}].

Then, for C^(K/2) √{square root over (D)}T sufficiently small comparedto 1, for any |F_(i)| in that interval, one has

$\begin{matrix}\left\langle {{{\exp\left( {{- {i\left( {{\frac{\alpha}{D}Z_{i}F_{i}} + X_{i}} \right)}}T} \right)}\psi_{+}{❘X_{i}❘}{\exp\left( {{- {i\left( {{\frac{\alpha}{D}Z_{i}F_{i}} + X_{i}} \right)}}T} \right)}\psi_{+}} \leq {1 - {\frac{\alpha^{2}T^{2}}{D}{{\exp\left( {- {O(K)}} \right)}.}}}} \right. & (28)\end{matrix}$

Thus, if this time-independent approximation is valid (and valid for alli), one has that

τ_(T)(H _(Z))

₊ ≥αT ² exp(−O(K))N.   (29)

Remark: here, an upper bound on force F_(i) was used because of thefixed time. If one averages over times on an interval, such an upperbound is not necessary.

B. Second Derivative of Expectation Value of X

One can give another approximate analysis by considering

$\begin{matrix}{{{\partial_{T}^{2}\left\langle {\tau_{T}^{H}\left( X_{i} \right)} \right\rangle} = {{{- \frac{\alpha^{2}}{D}}\left\langle {\tau_{T}^{H}\left( {X_{i}F_{i}^{2}} \right)} \right\rangle_{+}} + {\frac{\alpha}{D}\left\langle {\tau_{T}^{H}\left( {Y_{i}{\overset{.}{F}}_{i}} \right)} \right\rangle_{+}}}},} & (30)\end{matrix}$where for any operator O, define {dot over (O)}=−i[O,H].

For T=0 the first term is equal to

$- {\frac{\alpha^{2}}{D}.}$Assuming (which assumption is considered in more detail below) that thefirst term remains

${{- {\Omega(1)}}\frac{\alpha^{2}}{D}},$then one has

$\left\langle {\tau_{T}^{H}\left( X_{i} \right)} \right\rangle = {1 - \frac{\alpha^{2}T^{2}}{D}}$unless the second term

$\frac{\alpha}{D}\left\langle {\tau_{T}^{H}\left( {Y_{i}{\overset{.}{F}}_{i}} \right)} \right\rangle_{+}$also becomes

$\Omega(1){\frac{\alpha^{2}}{D}.}$For this to happen, one needs

τ_(T) ^(H)(Y_(i){dot over (F)}_(i))

₊=Ω(1)α.

Thus, under the assumption about the first term, one has one of twosituations. Either, after time T, one has

$\left\langle {\tau_{T}^{H}(X)} \right\rangle = {N \cdot \left( {1 - {{\Omega(1)}\frac{\alpha^{2}T^{2}}{D}}} \right)}$so that

$\left\langle {\tau_{T}^{H}\left( H_{Z} \right)} \right\rangle = {{\Omega(1)}\frac{\alpha^{2}T^{2}}{D}}$or one has Σ_(i)

τs^(H)(Y_(i){dot over (F)}_(i))

_(+=Ω()1)αN for some time s≤T. Further, at that time s, if one does nothave

$\left\langle {{\tau_{T}^{H}(X)} \geq {N \cdot \left( {1 - {{O(1)}\frac{\alpha^{2}T^{2}}{D}}} \right)}} \right.$then one has

$\left\langle {\tau_{T}^{H}\left( H_{Z} \right)} \right\rangle = {{\Omega(1)}{\frac{\alpha\; T^{2}}{D}.}}$

So, either the algorithm finds a state (by sampling over times s≤T) withexpectation value of H_(Z) equal to

${\Omega(1)}\frac{\alpha\; T^{2}}{D}$or for some state with expectation value of X at least

$N \cdot \left( {1 - {{O(1)}\frac{\alpha^{2}T^{2}}{D}}} \right)$and expectation value of Σ_(i)Y_(i){dot over (F)}_(i) at least Ω(1)α.Choosing α²T²˜D, this yields the same guarantees as the classicalalgorithm for α=√{square root over (D)}/ϵ, since using the same roundingas in the classical case, a large expectation value of Σ_(i)Y_(i){dotover (F)}_(i) implies one can construct a solution with a large absolutevalue of expectation value of H_(Z).

For smaller α²T², one will see that additional guarantees can be given.

C. Toy Example

The Hamiltonian of Eq. (3) provides an interesting example to study thetime-dependence of the force. Defining Z=Σ_(i)Z_(i), for the Hamiltonian(3) one has H_(Z)=−½Z²+const. (This constant is negative and of orderN.) Hence, up to an additive constant, one has

${H = {X = {{\frac{\alpha}{2D}Z^{2}} \approx {X - {\frac{\alpha}{2N}Z^{2}}}}}},$since D=N−1.This system can be approximately treated as a harmonic oscillator, atleast for X close to N. One can work in all eigenbasis of Z, lettingstate |z

denote an eigenstate of Z with eigenvalue z. In the large X regime, thewavefunction has most of its probability on basis states with i close tozero where the X operator is approximately equal to (N/2)|z

z+1|+h.c.. One can approximate further by treating z as a continuousvariable, approximating (N/2)|z

z+1|+h.c. by N+(N/2)∂_(z) ², valid in the long wavelength regime. Onethen gets that the Hamiltonian is equal to (ignoring additive constants)approximately equal to

$\frac{N}{2}{\partial_{z}^{2}{- \frac{\alpha}{2N}}}z^{2}$

Other than having a minus sign in front, this Hamiltonian is thefamiliar Hamiltonian for a harmonic oscillator. The oscillator hasangular frequencyω=√{square root over (α)}.   (31)

The z variable oscillates periodically with time at the given frequency.The force F_(i) at time t is (in this continuum approximation) equal toz(t). Hence, if αT²≳1, then the time-dependence of the force cannot beneglected in this example. Note that here again, one can see thisproduct αT² appearing; in the time-independent analysis above (and inthe previous heuristic analysis), this product controls the expectationvalue of H_(Z). Thus, it is no surprise that for this toy example thetime-independent approximation breaks down since there is no way to makethe expectation value of H_(Z) be large compared to N for this instance.

D. Time-Dependent Force

DefineΔ_(s) =F _(i)−τ_(s) ^(H)(F _(i)).   (32)One has (this hold for any state on the right-hand side, if one replacesψ₊ with an arbitrary state in the next two equations):

$\begin{matrix}{{{{\exp\left( {- {iHT}} \right)}{\exp\left( {{- i}\frac{\alpha}{D}{\int_{0}^{T}{Z_{i}\Delta_{s}{ds}}}} \right)}\psi_{+}} = {{{\exp\left( {- {iHT}} \right)}\psi_{+}} + {{\exp\left( {- {iHT}} \right)}\xi}}},} & (33)\end{matrix}$where the exponential is an s-ordered exponential (e.g., it istime-ordered with respected to s, are are later exponent of integralsbelow)

$\begin{matrix}{\xi = {{- i}\frac{\alpha}{D}{\int_{0}^{T}{{ds}\mspace{14mu}{\exp\left( {i - {\frac{\alpha}{D}{\int_{s}^{T}{\Delta_{u}{du}}}}} \right)}Z_{i}\Delta_{s}{\psi_{+}.}}}}} & (34)\end{matrix}$So, since

$\exp\left( {i - {\frac{\alpha}{D}{\int_{s}^{T}{\Delta_{u}{du}}}}} \right)$is unitary, by a triangle inequality, on has

$\begin{matrix}{{\xi } \leq {\frac{\alpha}{D}{\int_{s}^{T}{{ds}{\sqrt{\left\langle \Delta_{s}^{2} \right\rangle_{+}}.}}}}} & (35)\end{matrix}$

Define

$\begin{matrix}{{\phi(T)} = {{\exp\left( {- {iHT}} \right)}{\exp\left( {{- i}\frac{\alpha}{D}{\int_{0}^{T}{Z_{i}\Delta_{s}{ds}}}} \right)}{\psi_{+}.}}} & (36)\end{matrix}$This definition of ϕ(T) has the following property as can be seen bygoing to the interaction representation.Define operator R byH=X _(i) +Z _(i) F _(i) +R,   (37)so that R includes all terms in H which are not supported on site i.Then,

${\phi(T)} = {{\exp\left( {- {iRT}} \right)}{\exp\left( {{- {i\left( {{\frac{\alpha}{D}Z_{i}F_{i}} + X_{i}} \right)}}T} \right)}{\psi_{+}.}}$Hence,

$\begin{matrix}{\left\langle {{\phi(T)}{X_{i}}{\phi(T)}} \right\rangle = \left\langle {{\exp\left( {{- {i\left( {{\frac{\alpha}{D}Z_{i}F_{i}} + X_{i}} \right)}}T} \right)}\psi_{+}{X_{i}}{\exp\left( {{- {i\left( {{\frac{\alpha}{D}Z_{i}F_{i}} + X_{i}} \right)}}T} \right)}{\psi_{+}.}} \right.} & (38)\end{matrix}$

Defineψ₊(T)=exp(−iTH)ψ₊.   (39)So,ϕ(T)=ψ₊(T)+ξ.  (40)

Hence,

ϕ(T)|X _(i)|ϕ(T)

=

τ_(T)(X _(i))

₊+2Re

ψ₊(T)|X _(i)|ξ

ξ|X _(i)|ξ

.  (41)Let Π_(i) ⁻=(1−X_(i))/2, so that it projects onto the |−

state on qubit i. So,

ϕ(T)|Π_(i) ⁻|ϕ(T)

=

τ_(T)(Π_(i) ⁻)

₊+2Re

ψ₊(T)|Π_(i) ⁻|ξ

+

ξ|Π_(i) ⁻|ξ

.   (42)By Cauchy-Schwarz, the second term in the above equation is bounded by2√{square root over (

τ_(T) ^(H)(Π_(i) ⁻)

₊)}|ξ|. The third term is bounded by |ξ|².

Hence,

ϕ(T)|Π_(i) ⁻|ϕ(T)

≤

τ_(T)(Π_(i) ⁻)

₊+2√{square root over (

τ_(T) ^(H)(Π_(i) ⁻)

₊)}|ξ|+|ξ|².   (43)So,

τ_(T) ^(H)(Π_(i) ⁻)

₊≥

ϕ(T)|Π_(i) ⁻|ϕ(T)

−2√{square root over (

τ_(T) ^(H)(Π_(i) ⁻)

₊)}|ξ|−|ξ|².   (44)Thus,

τ_(T) ^(H)(Π_(i) ⁻)

₊≥

ϕ(T)|Π_(i) ⁻|ϕ(T)

−2√{square root over (

ϕ(T)|Π_(i) ⁻|ϕ(T)

)}|ξ|−|ξ|².   (45)

So, if one can bound |ξ| sufficiently small compared to √{square rootover (

ϕ(T)|Π_(i) ⁻|ϕ(T)

)}, then one can lower bound

τ_(T) ^(H)(Π_(i) ⁻)

₊ compared to

ϕ(T)|Π_(i) ⁻|ϕ(T)

. For example, if one can bound that |ξ|≤√{square root over (

ϕ(T)|Π_(i) ⁻|ϕ(T)

)}/3, then

τ_(T) ^(H)(Π_(i) ⁻)

₊≥

ϕ(T)|Π_(i) ⁻|ϕ(T)

·(1−2/3−1/9)=(2/9)·

ϕ(T)|Π_(i) ⁻|ϕ(T)

. If one can give even tighter bounds on then |ξ|, then

τ_(T) ^(H)(Π_(i) ⁻)

₊→

ϕ(T)|Π_(i) ⁻|ϕ(T)

as |ξ|→0.

So, one can now bound |ξ|². From Eq. (35),

${\xi } \leq {\frac{\alpha}{D}{\int_{0}^{T}{{ds}\mspace{14mu}{\sqrt{\left\langle \Delta_{s}^{2} \right\rangle_{+}}.}}}}$So, one turns to bounding

Δ_(s) ²

₊.So, Δ_(s)ψ₊=−∫₀ ^(s) dvτ_(v) ^(H) ({dot over (F)}_(i)), since Δ₀=0. So,again by Cauchy-Schwarz

Δ_(s) ²

+≤s ∫₀ ^(s) dv

|τ_(v) ^(H)({dot over (F)}_(i))|²

₊.   (46)So,

$\begin{matrix}{{\xi } \leq {\frac{\alpha}{D}{\int_{0}^{T}{{ds}\mspace{14mu} s{\sqrt{\frac{\int_{0}^{s}{d\;\upsilon\left\langle {{\tau_{\upsilon}^{H}\left( {\overset{.}{F}}_{i} \right)}}^{2} \right\rangle_{+}}}{s}}.}}}}} & (47)\end{matrix}$

So, |ξ| is bounded by αT²/(2D) times the expectation value of √{squareroot over (

|τ_(v) ^(H)({dot over (F)}_(i))|²

₊)} for s randomly chosen in the interval [0,T] from measure (T²/2)⁻ sdsand v uniformly randomly chosen in the interval [0, s] This randomchoice of s followed by a random choice of v induces a measuredμ(v)=2(1−v)dv.   (48)

Thus, to have

τ_(T) ^(H)(Π_(i) ⁻)

₊/

ϕ(T)|Π_(i) ⁻|ϕ(T)

small compared to 1, given that

ϕ(T)|Π_(i) ⁻|ϕ(T)

is at least exp(−O(K))(α²T²)/D as shown in subsection VIA, then oneneeds that for random choice of v from the measure μ(v), that theexpectation value

_(v)[√{square root over (

τ_(v) ^(H)({dot over (F)}_(i))|²

₊])} is at least

$\begin{matrix}{{{\exp\left( {- {O(K)}} \right)}\sqrt{\frac{\alpha^{2}T^{2}}{D}}\frac{2D}{\alpha\; T^{2}}} = {{\exp\left( {- {O(K)}} \right)}\frac{\sqrt{D}T}{\cdot}}} & (49)\end{matrix}$

Intuitively, Eq. (49) is clear: the magnitude of {dot over (F)}_(i),e.g., “how quickly the force is changing in time”, must be comparable tothe force at time 0 (e.g., to √{square root over (D)}) divided by thetime T, in order for the force to be small at time T.

Hence, the following lemma can be developed:

Lemma 4. For T≤exp(−O(K))/√{square root over (D)}, at least one of thefollowing two possibilities holds:

1.

τ_(T)(H _(Z))

₊ ≥αT ² exp(−O(K))N.   (50)

2. Σ_(i)

_(v)[√{square root over (

|τ_(v) ^(H)({dot over (F)}_(i))|²

₊])} is at least

${\exp\left( {- {O(K)}} \right)}{\frac{\sqrt{D}}{T}.}$

Proof. We have shown above that for each site i, if

τ_(T) ^(H)(Π_(i) ⁻)

₊/

ϕ(T)|Π_(i) ⁻|ϕ(T)) is compared to 1, then for random choice of v fromthe measure μ(v), that the expectation value

_(v)[√{square root over (

|τ_(v) ^(H)({dot over (F)}_(i))|²

₊])} is at least

${\exp\left( {- {O(K)}} \right)}{\frac{\sqrt{D}}{T}.}$If

τ_(T) ^(H)(Π_(i) ⁻)

₊/

ϕ(T)|Π_(i) ⁻|ϕ(T)

is compared to 1 for at least half the site i, then item 2. holds whileif it is small compared to 1 for fewer than half the sites then item 1.holds. □

At the time v of item 2 of lemma 4, at least one of the following holds:

⟨τ_(υ)(H_(Z))⟩ ≥ α T²exp (−O(K))N  or$\left\langle {\tau_{\upsilon}(X)} \right\rangle \geq {\left( {1 - {\frac{\alpha^{2}T^{2}}{D}{\exp\left( {- {O(K)}} \right)}}} \right) \cdot {N.}}$Hence, considering the mixed state averaged over v, these results holdin expectation.

Hence

Theorem 2. For T≤exp(−O(K))/√{square root over (D)}, at least one of thefollowing two possibilities holds: for some s ∈[0,T] one has that

1. Using a quench, one can produce a quantum state in polynomial timeswith expectation value of H_(Z)≥αT²exp(−O(K))N.

2. For some state ψ with

$\left\langle {\psi{X}\psi} \right\rangle \geq {\left( {1 - {\frac{\alpha^{2}T^{2}}{D}{\exp\left( {- {O(K)}} \right)}}} \right).}$N, one has that

${\sum\limits_{i}\sqrt{\left\langle {\psi{{{\overset{.}{F}}_{i}}^{2}}\psi} \right\rangle}} \geq {{\exp\left( {- {O(K)}} \right)}\frac{\sqrt{D}T}{\cdot}}$Proof. If item 1. of lemma 4 holds, then item 1. of this theorem hold bychoosing s=T. If item 2. of lemma 4 holds, and

${E_{\upsilon}\left\lbrack {\left\langle {\tau_{\upsilon}(X)} \right\rangle < {\left( {1 - {\frac{\alpha^{2}T^{2}}{D}{\exp\left( {- {O(K)}} \right)}}} \right) \cdot N}} \right\rbrack},$then E_(v)[

τ_(v)(H_(Z))

]≥αT² exp(−O(K))N and so by choosing a random v, item 1. of this theoremholds by averaging over time v. If item 2. of lemma 4 holds and

${E_{\upsilon}\left\lbrack {\left\langle {\tau_{\upsilon}(X)} \right\rangle \geq {\left( {1 - {\frac{\alpha^{2}T^{2}}{D}{\exp\left( {- {O(K)}} \right)}}} \right) \cdot N}} \right\rbrack},$then the mixed state averaged over time t obeys the conditions of item2. and so some pure state will obey the conditions also. □

This state ψ of item 2 has the property that

${\sum\limits_{i}\sqrt{\left\langle {\psi{{{\overset{.}{F}}_{i}}^{2}}\psi} \right\rangle}} \geq {{\exp\left( {- {O(K)}} \right)}{\frac{\sqrt{D}}{T}.}}$One can use this state ψ to construct another state which has a largeexpectation value for the absolute value of H_(Z); similar to theclassical rounding, this large expectation value may be very negativerather than very positive. An additional ingredient, discussed insubsection VIE is the interesting regime with a large expectation valueof X where additional bounds can be derived.

To do this, one can pick a random set of sites, called S. One caninclude each site in this set with probability ½, choosing independentlyfor each site whether or not it is in S. Note that {dot over (F)}_(i) isa degree-(K−1) polynomial. Each term in the polynomial is degree K−2 inPauli Z-variable and degree 1 in Pauli Y variables. Define F_(i) ^(S) toinclude only the terms in the polynomial which do not include sites inS. The basic idea (we will give this in more detail below) is to chooseY_(i) for i ∈S to be proportional to F_(i) ^(S) ; with the constant ofproportionality chosen to have

$\left\langle X_{i} \right\rangle \geq {1 - {\frac{\alpha^{2}T^{2}}{2D}{\exp\left( {- {O(K)}} \right)}}}$for i ∈S.

In expectation over choices of S, one has

${\sum_{i \in S}\sqrt{\left\langle {\psi{{{\overset{.}{F}}_{i}^{\overset{\_}{S}}}^{2}}\psi} \right\rangle}} \geq {{\exp\left( {- {O(K)}} \right)}{\frac{\sqrt{D}}{T}.}}$So, indeed for some choice of S this holds. Let ρ be the mixed stateobtained by tracing out sites in S for such a choice. Let σ_(S) ⁺ be thedensity matrix on S with all spins polarized in the + direction.Consider the state

${\tau \equiv {{\exp\left( {{ic}{\sum\limits_{i \in S}{{\overset{.}{F}}_{i}^{\overset{\_}{S}}Z_{i}}}} \right)}\left( {\rho \otimes \sigma_{S}^{+}} \right){\exp\left( {{- {ic}}{\sum\limits_{i \in S}{{\overset{.}{F}}_{i}^{\overset{\_}{S}}Z_{i}}}} \right)}}},$where c is a constant chosen below.

So long as c{dot over (F)}_(i) ^(S) =O(1), one can find thattr(τX_(i))≥1−O(c²)

(ρ|F_(i) ^(S) |²) and that tr(τΣ_(i∈S)≥ctr(ρ|F_(i) ^(S) |²). Suppose forthe moment that

${\sum_{i \in S}\sqrt{\left\langle {\psi{{{\overset{.}{F}}_{i}^{\overset{\_}{S}}}^{2}}\psi} \right\rangle}} = {{\exp\left( {- {O(K)}} \right)}{\frac{\sqrt{D}}{T}.}}$That is, a lower bound on the left-hand side was previously assumed, butnow one can assume equality. Then, for c=(αT²)/D, one has

$\begin{matrix}{{{tr}\left( {\tau{\sum\limits_{i \in \mathcal{S}}{Y_{i}{\overset{.}{F}}_{i}^{\overset{\_}{\mathcal{S}}}}}} \right)} \geq {\alpha\;{{\exp\left( {- {O(K)}} \right)}.}}} & (51)\end{matrix}$Also,

$\begin{matrix}{{{tr}\left( {\tau\; X_{i}} \right)} \geq {1 - {{O\left( \frac{\alpha^{2}T^{2}}{D} \right)}.}}} & (52)\end{matrix}$Now, if it turns out that instead that Σ_(i∈S)√{square root over (

ψ||{dot over (F)}_(i) ^(S) |²|ψ

)} is significantly larger than

${{\exp\left( {{- O}(K)} \right)}\frac{\sqrt{D}}{T}},$one can instead reduce c proportionally and still obtain a state obeyingEqs. (51,52).

So, one finds that

Theorem 3. For T≤exp(−O(K))/√{square root over (D)}, at least one of thefollowing two possibilities holds: for some s ∈[0,T] one has that

1. Using a quench one can produce a quantum state in polynomial timeswith expectation value of H_(Z)≥αT² exp((−O(K))N.

2. For some state ψ with

$\left\langle {\psi{X}\psi} \right\rangle \geq {\left( {1 - {\frac{\alpha^{2}T^{2}}{D}{\exp\left( {- {O(K)}} \right)}}} \right).}$N, one has that

${{\sum\limits_{i \in \mathcal{S}}\left\langle {\psi{{X_{i}{\overset{.}{F}}_{i}}}\psi} \right\rangle}} \geq {{\exp\left( {{- O}(K)} \right)}\alpha\;{N.}}$

For the case α²T²/D=1, applying the same rounding as in the classicalcase to item 2, one finds the same duality as in the classical case, forαT²=ϵ√{square root over (D)} and α=√{square root over (D)}/ϵ.

In the next subsection, additional guarantees present in the quantumalgorithm when α²T²/D<<1 are explored.

Further, one can consider the expectation value of higher moments ofτ_(v)(X). The reason for considering this is explained later. The timeevolution conserves the quantity

${H + X + {\frac{\alpha}{D}H_{Z}}},$but it also conserves all moments of this quantity. Note that in thestate ψ₊ one has

(H−N)²

₊=(α²/D²)

H_(Z) ²

₊=(α²/D²)N_(T)=Nα²/(DK). Hence,

τ_(T) ^(H)(H−N²)

₊=Nα²/(DK). By Cauchy-Schwarz,

$\begin{matrix}{\left\langle {\tau_{T}^{H}\left( \left( {H - N} \right)^{2} \right)} \right\rangle_{+} = {{\left\langle {\tau_{T}^{H}\left( \left( {X - N} \right)^{2} \right)} \right\rangle_{+} + {2\frac{\alpha}{D}\left\langle {\tau_{T}^{H}\left( {\left( {X - N} \right)H_{Z}} \right)} \right\rangle_{+}\frac{\alpha^{2}}{D^{2}}\left\langle {\tau_{T}^{H}\left( H_{Z}^{2} \right)} \right\rangle_{+}}} \geq {\left\langle {\tau_{T}^{H}\left( \left( {X - N} \right)^{2} \right)} \right\rangle_{+} + \left\langle {\tau_{T}^{H}\left( H_{Z}^{2} \right)} \right\rangle_{+} - {2{\sqrt{\left\langle {\tau_{T}^{H}\left( H_{Z}^{2} \right)} \right\rangle_{+}\left\langle {\tau_{T}^{H}\left( \left( {X - N} \right)^{2} \right)} \right\rangle_{+}}.}}}}} & (53)\end{matrix}$Hence,

$\begin{matrix}{\sqrt{\left\langle {\tau_{T}^{H}\left( H_{Z}^{2} \right)} \right\rangle_{+}} \geq {{\frac{D}{\alpha}\sqrt{\left\langle {\tau_{T}^{H}\left( \left( {X - N} \right)^{2} \right)} \right\rangle_{+}}} + {\sqrt{N_{T}}.}}} & (54)\end{matrix}$

Hence, one has related flucutations in X−N to fluctuations in H. If itis the case that with probability at most (αT²/D)² that τ_(T)^(H)(H_(Z)) is measured to be greater than αT²N, then since ∥H_(Z)∥≤DN,it follows that √{square root over (

τ_(T) ^(H)(H_(Z) ²)

₊)}=O(αT²N), and so

$\begin{matrix}{\sqrt{\left\langle {\tau_{T}^{H}\left( \left( {X - N} \right)^{2} \right)} \right\rangle_{+}} = {{O\left( {{\frac{\alpha^{2}T^{2}}{D}N} + {\frac{\alpha}{D}\sqrt{N_{T}}}} \right)}.}} & (55)\end{matrix}$In the limit of large N, the quantity √{square root over (N_(T))} isasymptotically only √{square root over (N)} and so is negligiblecompared to the leading term.

E. Large X Expectation Value in Duality

The quantum algorithm when α²T²/D<<1 is now considered, e.g., when theexpectation value of X in item 2 is close to 1.

First, a simple mean-field treatment is given: consider some Hamiltonianof degree K that will be called H₀ that is diagonal in the Z-basis.Suppose one wishes to maximize the expectation value of H₀ over stateswith given expectation value of X. If no constraint were placed on theexpectation value of X, then one can maximize H_(Z) by choosing somestate in the computational basis. For each spin i, this state has someexpectation value

Z_(i)

=z_(i) with z_(i)∈{−1, +1}. If one wishes to obtain a nonzeroexpectation value of X, then a simple way is to take a product state,where each spin has z,142 X_(i)

=cos(θ) and

Z_(i)

=z_(i) sin(θ), for some angle θ. For θ=π/2, one can recover theclassical state. At small θ, the expectation value of H₀ is proportionalto θ^(K), while the expectation value of 1−X_(i) is proportional to θ².Thus, for K>2, the expectation value of H₀ drops more rapidly as afunction of θ than does the expectation value of 1−X_(i).

A similar mean-field treatment might be applied to a Hamiltonian H₀ thatincludes both Y and Z operators: given any product solution of H₀ with

Z_(i)

=z_(i) and

Y_(i)

=y_(i) with z_(i) ²+y_(i) ²=1, one can define a product state with

X_(i)

=cos(θ) and

Z_(i)

=z_(i) sin(θ) and

Y_(i)

=y_(i) sin(θ).

If this mean-field procedure were the best possible then, combined withtheorem 3 taking H₀ to be the Hamiltonian of item 2, one would have avery favorable situation for the quantum algorithm: one would have (forsmall θ) the scaling θ²˜α²T²/D and while the expectation value of H₀would be at most θ^(K) times the optimal value of H₀. Call this optimalvalue H₀ ^(max). Then for case 2 to apply, one would need α˜θ^(K)H₀^(max) while α²T²/D˜θ². So one would have (αT²)˜Dθ^(2−K)/H₀ ^(max).Taking, at the most optimistic situation, θ˜1/√{square root over (D)}(since for smaller θ the expectation value of X_(i) is within 1/D of 1and certainly the mean-field is not accurate here), one would find thatcase 1 holds unless H₀ ^(max)˜N(αT²)⁻¹D^(K/2). For the case K=2, this isthe same guarantee as before, but for K=4 or larger, this is muchstronger guarantee.

This mean-field procedure will break down, so one cannot give such astrong guarantee for the quantum algorithm. However, in this subsection,it will be shown that some guarantees are still present in the quantumalgorithm. These guarantees are expressed in terms of a semi-definiteprogramming optimization problem.

Here, the case K=4 for definiteness is fixed from now on. Also, considerfrom here on the dense case, where N_(T)˜N^(K). The semi-definiteprogramming problem that is given will still be relevant even if thedense case is not considered; however, in the dense case some additionalinteresting bounds can be given.

The dense case was studied previously, where it was shown that one canin general improve upon a random assignment by an amount proportional to√{square root over (N_(T))}. For K=4, this means that one can achieve

H^(Z)

˜N² in the worst case. This is interesting as the problem has degreeD˜N³ and so the improvement over random even in the worst case is bymuch more than N_(T)/D.

In fact, the classical algorithm above will typically also achieve avalue at least proportional to N² for all instances. To see this, notethat in case 3 of theorem 1, the polynomial p(x) in the proof will havecoefficient a₁ of order N√{square root over (D)} but the second ordercoefficient a₂ will be much smaller than N D. Indeed, for any given pairof sites i, j ∈S, the coefficient of ({right arrow over(w)}₁)_(i)({right arrow over (w)}₁)_(j) will be a weighted sum of terms({right arrow over (w)}₂)_(k)({right arrow over (w)}₂)_(t) for k, l ∉S.While there are N² terms in the sum, for random choice of {right arrowover (w)}₂, the magnitude of this term will typically only be of orderN. So, one will typically have a₂ of order N². Similar bounds can begiven on higher degree coefficients of the poynomial. So, for x˜a₁/|a₂|,one will have p(x)˜a₁ ²/a₂˜N D/N²˜N² as claimed.

The quantum algorithm will also typically achieve a value at leastproportional to N² for all instances. To see this, consider anyHamiltonian H₀ which is a sum of terms of degree K, all with coefficient+1. Assume that H₀ is diagonal in the Z basis (a similar calculationarises for any H₀ which includes terms in both Y and Z basis). It isdesirable to optimize H₀ at given expectation value of X. One can claimthat for any such choice of H₀, the optimum is bounded by the optimumwhere all coefficients in H₀ are +1. To see this, work in an eigenbasisof X_(i). Then, H₀ becomes an off-diagonal operator; for anywavefunction optimizing any given choice of H₀, if one takes theabsolute value of that wavefunction (e.g., in the eigenbasis of X_(i),one replaces each coefficient with its absolute value), one obtains atleast as large an eigenvalue for the case where all coefficients are +1.Indeed, one obtains at least as large a value if one assumes that allcoefficients are +1 and non-vanishing. However, if all coefficients inH₀ are +1, this is a soluble problem: one must optimize X+Z⁴. Optimizingthis when the expectation value of X is N−1, one finds that the optimumis ˜N². This indeed is what one expects from mean-field theory for thisproblem. Note that here rather than taking expectation value of1−X_(i)˜1/D, one instead takes the expectation value of 1−X_(i)˜1/N inthe dense case.

At this point a slight change of notation makes things clearer in thedense case. Let one define β so that

$\frac{\beta}{\sqrt{N_{T}}} = {\frac{\alpha}{D}.}$

Then, one can find that case 2 of theorem 2 has expectation value N−X oforder β²T², while the expectation value of H₀ is of order β√{square rootover (N_(T))} and in case 1 the expectation value of H_(Z) is of orderβT²√{square root over (N_(T))}. Choosing β²T²=1 and choosing β slightlylarger than 1, one sees that case 2 cannot then occur so case 1 mustoccur so indeed one gets an expectation value of H_(Z) of order √{squareroot over (N_(T))}.

Now, more general instances are considered and a bound is given on theexpectation value of H₀ in general.

First, the case of K=2 is considered simply to fix notation and thegeneral framework is given. The case of k=4 is then considered. Itshould be understood that generalization to higher cases is possible andwithin the scope of the disclosed technology.

For each site i, define operator b_(i) ^(†)=(|−

+|)_(i). That is, it has a nonzero matrix element only from the |+

state on i to the |−)

state. One has b_(i)b_(i) ^(†)=(1+X_(i))2 and b_(i)b_(i)^(†)=(1−X_(i))/2. For any quantum state (possibly mixed) ρ define a2N-by-2N matrix M of correlation functions. This matrix will have ablock form

$\begin{matrix}{{M = \begin{pmatrix}M_{++} & M_{+ -} \\M_{- +} & M_{--}\end{pmatrix}},} & (56)\end{matrix}$where M₊₊ has matrix elements(M ₊₊)_(ij) =tr(ρb _(i) ^(†) b _(j)),   (57)and M⁺⁻ has matrix elements(M ⁺⁻)_(ij) =tr(ρb _(i) b _(j)),   (58)M⁻⁻ has matrix elements(M ⁻⁻)_(ij) =tr(ρb _(i) b _(j) ^(†)).   (59)One can set M⁺⁻=M⁻⁺ ^(†) and the matrix M₊₊ and M⁻⁻ are Hermitian. Notethen that the off-diagonal elements of M₊₊ and M⁻⁻ are related by(M ₊₊)_(ij)=⁻(M ⁻⁻)_(ij)for i≠j but(M ₊₊)_(ii)=1−(M ⁻⁻)_(i) i.

Equivalently, given an 2N component vector {right arrow over (a)}, onecan define an operator O({right arrow over (a)}) by

${O\left( \overset{\rightarrow}{a} \right)} = {{\sum\limits_{i = 1}^{N}{b_{i}^{\dagger}a_{i}}} + {\sum\limits_{i = 1}^{N}{b_{i}{a_{i + N}.}}}}$

Then, M is such thattr(ρO({right arrow over (a)} ₁)^(†) O({right arrow over (a)} ₂))={rightarrow over (a)} ₁ ^(†) ·M·{right arrow over (a)} ₂.   (60)So, M is the matrix of correlations functions of a 2N component vectorcontaining operators b^(†) and b.

Then, since for any {right arrow over (a)} and any ρ, one hastr(ρO({right arrow over (a)})^(†)O({right arrow over (a)}))≥0, itfollows that M is a positive semi-definite matrix. Further, one has

$\begin{matrix}{{{tr}\left( M_{++} \right)} = {{{tr}\left( {\rho\frac{N - X}{2}} \right)}.}} & (61)\end{matrix}$

One can construct a similar semi-definite programming bound in the caseK=4 or larger. Now one can construct a matrix M which is a(2N)^(K/2)-by-(2N)^(K/2)) matrix. Let one focus on the case K=4. Todefine this matrix M, given any (2N)² component vector {right arrow over(a)}, label the components by a pair (i, j) each ranging from 1 to 2N.Then define

${O\left( \overset{\rightarrow}{a} \right)} = {\sum\limits_{i = 1}^{N}{\sum\limits_{j = 1}^{N}{\left( {{{\overset{\rightarrow}{a}}_{i,j}b_{i}^{\dagger}b_{j}^{\dagger}} + {{\overset{\rightarrow}{a}}_{{i + N},j}b_{i}^{\dagger}b_{j}^{\dagger}} + {{\overset{\rightarrow}{a}}_{i,{j + N}}b_{i}^{\dagger}b_{j}^{\dagger}} + {{\overset{\rightarrow}{a}}_{{i + N},{j + N}}b_{i}^{\dagger}b_{j}^{\dagger}}} \right).}}}$

Again the matrix M is positive semi-definite. One can again write M witha block structure

$\begin{matrix}{M = {\begin{pmatrix}M_{++{;++}} & M_{++{;{+ -}}} & M_{++{;{- +}}} & M_{++{;--}} \\M_{{+ -};++} & M_{{+ -};{+ -}} & M_{{+ -};{- +}} & M_{{+ -};--} \\M_{{- +};++} & M_{{- +};{+ -}} & M_{{- +};{- +}} & M_{{- +};--} \\M_{--{;++}} & M_{--{;{+ -}}} & M_{--{;{- +}}} & M_{--{;--}}\end{pmatrix}.}} & (62)\end{matrix}$

The diagonal entries of M can be bounded in terms of the second momentof N−X This gives a semi-definite programming relaxation that allows oneto bound the expectation value of H₀ at large X.

VII. Discussion

The embodiments disclosed include an algorithm that uses quantumquenches as well as a classical algorithm to perform approximateoptimization. It was also proven that both of these algorithms obtain aresult that improves upon the random appraoch by an amount that is morethan N/D unless a related problem has a “very bad” solution. This can beused then in some cases to guarantee that the algorithm will find anontrivial improvement if no such solution of the related problemexists. Additional guarantees can be given for the quantum algorithm.

The example quench algorithm is not described by a fixed depth quantumcircuit, independent of D. The Lieb-Robinson velocity v_(LR) of thisHamiltonian is proportional to √{square root over (α)}, as can be shownby using Lieb-Robinson bounds adapted to Hamiltonians where theHamiltonians is a sum of two types of terms (in this case, X_(j) fordifferent qubits j is one type and terms in H_(Z) is another type) suchthat terms within a type commute; more generally, one can use boundsadapted to the case of a bounded commutator. To define the Lieb-Robinsonvelocity, one can define a distance between qubits by using a graphmetric for a graph with vertices corresponding to qubits and an edgebetween vertices if the corresponding qubits are both in some term inH_(Z).

The estimates using the Lieb-Robinson velocity give some upper bound onhow far a perturbation can propagate in a given time; the effect of aperturbation beyond a distance proportional to v_(LR)t is negligible.These estimates may not be tight, but it is to be expected that indeedthe velocity of perturbations will be proportional to √{square root over(α)} in many systems. If this is true, then if αt² diverges with D toobtain a nontrivial approximation, the necessary circuit depth alsodiverges.

VIII. General Embodiments

In this section, example methods for performing aspects of the disclosedembodiments are disclosed. The particular embodiments described shouldnot be construed as limiting, as the disclosed method acts can beperformed alone, in different orders, or at least partiallysimultaneously with one another. Further, any of the disclosed methodsor method acts can be performed with any other methods or method actsdisclosed herein.

FIG. 5 is an example method for performing an approximate optimizationtechnique using a quantum quench algorithm as disclosed herein.

At 510, a quantum computing device is configured to perform anapproximate optimization technique to approximate a solution to acombinatorial optimization problem.

At 512, the approximate optimization technique is performed on thequantum computing device, wherein the approximate optimization techniqueincludes using a quantum quench algorithm.

In certain implementations, the quench algorithm includes averagingstate values over a plurality of times. In particular implementations,the controlling comprises changing coupling constants without using ajump or slow change in the coupling constants. In some examples, thechanging of the coupling constants is performed non-adiabatically. Incertain examples, the changing of the coupling constants is followed byan equilibration time.

In some implementations, the method further comprises reading outresults of the approximate optimization technique from the quantumcomputing device; and storing the results in a classical computingdevice.

FIG. 6 is another example method for performing an approximateoptimization technique using a quantum quench algorithm as disclosedherein.

At 610, a quantum quench algorithm is performed on a classical computingdevice using simulation of a quantum Hamiltonian to perform approximateoptimization. In certain implementations, the quench algorithm includesaveraging state values over a plurality of times.

IX. Example Computing Environments

FIG. 1 illustrates a generalized example of a suitable classicalcomputing environment 100 in which several of the described embodimentscan be implemented. The computing environment 100 is not intended tosuggest any limitation as to the scope of use or functionality of thedisclosed technology, as the techniques and tools described herein canbe implemented in diverse general-purpose or special-purposeenvironments that have computing hardware.

With reference to FIG. 1 , the computing environment 100 includes atleast one processing device 110 and memory 120. In FIG. 1 , this mostbasic configuration 130 is included within a dashed line. The processingdevice 110 (e.g., a CPU or microprocessor) executes computer-executableinstructions. In a multi-processing system, multiple processing devicesexecute computer-executable instructions to increase processing power.The memory 120 may be volatile memory (e.g., registers, cache, RAM,DRAM, SRAM), non-volatile memory (e.g., ROM, EEPROM, flash memory), orsome combination of the two. The memory 120 stores software 180implementing tools for peforming any of the approximation methodsdisclosed herein for addressing combinatorial optimization problems,using a classical computer and/or quantum computer. For example, thememory 120 can store software for controlling a quantum circuit toimplement an embodiment of the disclosed approximation technique. Thememory 120 can also store software 180 for synthesizing, generating (orcompiling), and/or controlling quantum circuits as described herein.

The computing environment can have additional features. For example, thecomputing environment 100 includes storage 140, one or more inputdevices 150, one or more output devices 160, and one or morecommunication connections 170. An interconnection mechanism (not shown),such as a bus, controller, or network, interconnects the components ofthe computing environment 100. Typically, operating system software (notshown) provides an operating environment for other software executing inthe computing environment 100, and coordinates activities of thecomponents of the computing environment 100.

The storage 140 can be removable or non-removable, and includes one ormore magnetic disks (e.g., hard drives), solid state drives (e.g., flashdrives), magnetic tapes or cassettes, CD-ROMs, DVDs, or any othertangible non-volatile storage medium which can be used to storeinformation and which can be accessed within the computing environment100. The storage 140 can also store instructions for the software 180implementing tools for peforming any of the approximation methodsdisclosed herein for addressing combinatorial optimization problems,using a classical computer and/or quantum computer. For example, thememory 120 can store software for controlling a quantum circuit toimplement an embodiment of the disclosed approximation technique. Thestorage 140 can also store instructions for the software 180 forsynthesizing, generating (or compiling), and/or controlling quantumcircuits as described herein.

The input device(s) 150 can be a touch input device such as a keyboard,touchscreen, mouse, pen, trackball, a voice input device, a scanningdevice, or another device that provides input to the computingenvironment 100. The output device(s) 160 can be a display device (e.g.,a computer monitor, laptop display, smartphone display, tablet display,netbook display, or touchscreen), printer, speaker, or another devicethat provides output from the computing environment 100.

The communication connection(s) 170 enable communication over acommunication medium to another computing entity. The communicationmedium conveys information such as computer-executable instructions orother data in a modulated data signal. A modulated data signal is asignal that has one or more of its characteristics set or changed insuch a manner as to encode information in the signal. By way of example,and not limitation, communication media include wired or wirelesstechniques implemented with an electrical, optical, RF, infrared,acoustic, or other carrier.

As noted, the various methods, quantum circuit control techqniues, orcompilation/synthesis techniques can be described in the general contextof computer readable instructions stored on one or morecomputer-readable media. Computer-readable media are any available media(e.g., memory or storage device) that can be accessed within or by acomputing environment. Computer-readable media include tangiblecomputer-readable memory or storage devices such as memory 120 and/orstorage 140, and do not include propagating carrier waves or signals perse (tangible computer-readable memory or storage devices do not includepropagating carrier waves or signals per se).

Various embodiments of the methods disclosed herein can also bedescribed in the general context of computer-executable instructions(such as those included in program modules) being executed in acomputing environment by a processor. Generally, program modules includeroutines, programs, libraries, objects, classes, components, datastructures, and so on, that perform particular tasks or implementparticular abstract data types. The functionality of the program modulesmay be combined or split between program modules as desired in variousembodiments. Computer-executable instructions for program modules may beexecuted within a local or distributed computing environment.

An example of a possible network topology 200 (e.g., a client-servernetwork) for implementing a system according to the disclosed technologyis depicted in FIG. 2 . Networked computing device 220 can be, forexample, a computer running a browser or other software connected to anetwork 212. The computing device 220 can have a computer architectureas shown in FIG. 1 and discussed above. The computing device 220 is notlimited to a traditional personal computer but can comprise othercomputing hardware configured to connect to and communicate with anetwork 212 (e.g., smart phones, laptop computers, tablet computers, orother mobile computing devices, servers, network devices, dedicateddevices, and the like). Further, the computing device 220 can comprisean FPGA or other programmable logic device. In the illustratedembodiment, the computing device 220 is configured to communicate with acomputing device 230 (e.g., a remote server, such as a server in a cloudcomputing environment) via a network 212. In the illustrated embodiment,the computing device 220 is configured to transmit input data to thecomputing device 230, and the computing device 230 is configured toimplement a quantum circuit control technique according to any of thedisclosed embodiments and/or a circuit generation orcompilation/synthesis methods for generating qunatum circuits for usewith any of the techniques disclosed herein. The computing device 230can output results to the computing device 220. Any of the data receivedfrom the computing device 230 can be stored or displayed on thecomputing device 220 (e.g., displayed as data on a graphical userinterface or web page at the computing devices 220). In the illustratedembodiment, the illustrated network 212 can be implemented as a LocalArea Network (LAN) using wired networking (e.g., the Ethernet IEEEstandard 802.3 or other appropriate standard) or wireless networking(e.g. one of the IEEE standards 802.11a, 802.11b, 802.11g, or 802.11n orother appropriate standard). Alternatively, at least part of the network212 can be the Internet or a similar public network and operate using anappropriate protocol (e.g., the HTTP protocol).

Another example of a possible network topology 300 (e.g., a distributedcomputing environment) for implementing a system according to thedisclosed technology is depicted in FIG. 3 . Networked computing device320 can be, for example, a computer running a browser or other softwareconnected to a network 312. The computing device 320 can have a computerarchitecture as shown in FIG. 1 and discussed above. In the illustratedembodiment, the computing device 320 is configured to communicate withmultiple computing devices 330, 331, 332 (e.g., remote servers or otherdistributed computing devices, such as one or more servers in a cloudcomputing environment) via the network 312. In the illustratedembodiment, each of the computing devices 330, 331, 332 in the computingenvironment 300 is used to perform at least a portion of a quantumcircuit control technique according to any of the disclosed embodimentsand/or a circuit generation or compilation/synthesis methods forgenerating quantum circuits for use with any of the techniques disclosedherein. In other words, the computing devices 330, 331, 332 form adistributed computing environment in which the quantum circuit controland/or generation/compilation/synthesis processes are shared acrossmultiple computing devices. The computing device 320 is configured totransmit input data to the computing devices 330, 331, 332, which areconfigured to distributively implement such as process, includingperformance of any of the disclosed methods or creation of any of thedisclosed circuits, and to provide results to the computing device 320.Any of the data received from the computing devices 330, 331, 332 can bestored or displayed on the computing device 320 (e.g., displayed as dataon a graphical user interface or web page at the computing devices 320).The illustrated network 312 can be any of the networks discussed abovewith respect to FIG. 2 .

With reference to FIG. 4 , an exemplary system for implementing thedisclosed technology includes computing environment 400. In computingenvironment 400, a compiled quantum computer circuit description(including quantum circuits for performing any of the disclosedapproximation techniques) can be used to program (or configure) one ormore quantum processing units such that the quantum processing unit(s)implement the circuit described by the quantum computer circuitdescription.

The environment 400 includes one or more quantum processing units 402and one or more readout device(s) 408. The quantum processing unit(s)execute quantum circuits that are precompiled and described by thequantum computer circuit description. The quantum processing unit(s) canbe one or more of, but are not limited to: (a) superconducting quantumcomputer; (b) an ion trap quantum computer; (c) a fault-tolerantarchitecture for quantum computing; and/or (d) a topological quantumarchitecture (e.g., a topological quantum computing device usingMajorana zero modes). The precompiled quantum circuits, including any ofthe disclosed circuits, can be sent into (or otherwise applied to) thequantum processing unit(s) via control lines 406 at the control ofquantum processor controller 420. The quantum processor controller (QPcontroller) 420 can operate in conjunction with a classical processor410 (e.g., having an architecture as described above with respect toFIG. 1 ) to implement the desired quantum computing process. In theillustrated example, the QP controller 420 further implements thedesired quantum computing process via one or more QP subcontrollers 404that are specially adapted to control a corresponding one of the quantumprocessor(s) 402. For instance, in one example, the quantum controller420 facilitates implementation of the compiled quantum circuit bysending instructions to one or more memories (e.g.; lower-temperaturememories), which then pass the instructions to low-temperature control.unit(s) (e.g., QP subcontroller(s) 404) that transmit, for instance,pulse sequences representing the gates to the quantum processing unit(s)402 for implementation. In other examples, the QP controller(s) 420 andQP subcontroller(s) 404 operate to provide appropriate magnetic fields,encoded operations, or other such control signals to the quantumprocessor(s) to implement the operations of the compiled quantumcomputer circuit description. The quantum controller(s) can furtherinteract with readout devices 408 to help control and implement thedesired quantum computing process (e.g., by reading or measuring outdata results from the quantum processing units once available, etc.)

With reference to FIG. 4 , compilation is the process of translating ahigh-level description of a quantum algorithm into a quantum computercircuit description comprising a sequence of quantum operations orgates, which can include the circuits as disclosed herein. Thecompilation can be performed by a compiler 422 using a classicalprocessor 410 (e.g., as shown in FIG. 1 ) of the environment 400 whichloads the high-level description from memory or storage devices 412 andstores the resulting quantum computer circuit description in the memoryor storage devices 412.

In other embodiments, compilation and/or verification can be performedremotely by a remote computer 460 (e.g., a computer having a computingenvironment as described above with respect to FIG. 1 ) which stores theresulting quantum computer circuit description in one or more memory orstorage devices 462 and transmits the quantum computer circuitdescription to the computing environment 400 for implementation in thequantum processing unit(s) 402. Still further, the remote computer 400can store the high-level description in the memory or storage devices462 and transmit the high-level description to the computing environment400 for compilation and use with the quantum processor(s). In any ofthese scenarios, results from the computation performed by the quantumprocessor(s) can be communicated to the remote computer after and/orduring the computation process. Still further, the remote computer cancommunicate with the QP controller(s) 420 such that the quantumcomputing process (including any compilation, verification, and QPcontrol procedures) can be remotely controlled by the remote computer400. In general, the remote computer 460 communicates with the QPcontroller(s) 420, compiler/synthesizer 422, and/or verification tool423 via communication connections 450.

In particular embodiments, the environment 400 can be a cloud computingenvironment, which provides the quantum processing resources of theenvironment 400 to one or more remote computers (such as remote computer460) over a suitable network (which can include the internet).

X. Concluding Remarks

Having described and illustrated the principles of the disclosedtechnology with reference to the illustrated embodiments, it will berecognized that the illustrated embodiments can be modified inarrangement and detail without departing from such principles. Forinstance, elements of the illustrated embodiments shown in software maybe implemented in hardware and vice-versa. Also, the technologies fromany example can be combined with the technologies described in any oneor more of the other examples. It will be appreciated that proceduresand functions such as those described with reference to the illustratedexamples can be implemented in a single hardware or software module, orseparate modules can be provided. The particular arrangements above areprovided for convenient illustration, and other arrangements can beused.

What is claimed is:
 1. A method, comprising: configuring a quantumcomputing device to perform an approximate optimization technique toapproximate a solution to a combinatorial optimization problem; andperforming the approximate optimization technique on the quantumcomputing device, wherein the approximate optimization techniqueincludes using a quantum quench algorithm.
 2. The method of claim 1,wherein the quench algorithm includes averaging state values over aplurality of times.
 3. The method of claim 1, wherein the approximateoptimization technique comprises: changing coupling constants withoutusing a jump or slow change in the coupling constants.
 4. The method ofclaim 3, wherein the changing coupling constants is performednon-adiabatically.
 5. The method of claim 3, wherein the changing of thecoupling constants is followed by an equilibration time.
 6. The methodof claim 1, further comprising: reading out results of the approximateoptimization technique from the quantum computing device; and storingthe results in a classical computing device.
 7. A system, comprising: aclassical computer; and a quantum computer, wherein the quantumcomputing device is configured to perform an approximate optimizationtechnique on the quantum computing device to approximate a solution to acombinatorial optimization problem, wherein the approximate optimizationtechnique includes using a quantum quench algorithm.
 8. The system ofclaim 7, wherein the quench algorithm includes averaging state valuesover a plurality of times.
 9. The system of claim 7, wherein theapproximate optimization technique comprises changing coupling constantswithout using a jump or slow change in the coupling constants.
 10. Thesystem of claim 9, wherein the changing coupling constants is performednon-adiabatically.
 11. The system of claim 9, wherein the changing ofthe coupling constants is followed by an equilibration time.
 12. Thesystem of claim 9, wherein the classical computer is further configuredto: read out results of the approximate optimization technique from thequantum computing device; and store the results in a classical computingdevice.
 13. One or more computer-readable media storingcomputer-executable instructions, which when executed by a classicalcomputing device, cause the classical computing device to perform amethod, the method comprising: configuring a quantum computing device toperform an approximate optimization technique to approximate a solutionto a combinatorial optimization problem; and performing the approximateoptimization technique on the quantum computing device, wherein theapproximate optimization technique includes using a quantum quenchalgorithm.
 14. The one or more computer-readable media of claim 13,wherein the quench algorithm includes averaging state values over aplurality of times.
 15. The one or more computer-readable media of claim13, wherein the approximate optimization technique comprises: changingcoupling constants without using a jump or slow change in the couplingconstants.
 16. The one or more computer-readable media of claim 15,wherein the changing coupling constants is performed non-adiabatically.17. The one or more computer-readable media of claim 15, wherein thechanging of the coupling constants is followed by an equilibration time.18. The one or more computer-readable media of claim 13, furthercomprising: reading out results of the approximate optimizationtechnique from the quantum computing device; and storing the results.19. A method, comprising performing, on a classical computing device, aquantum quench algorithm using simulation of a quantum Hamiltonian toperform approximate optimization, wherein the quench algorithm includesaveraging state values over a plurality of times.